IEEE - Aerospace and Electronic Systems - March 2021 - 41

Feng et al.
Z
Sðk; fÞ ¼

T

s2 ðk; tÞeÀj2pft dt:

(10)

0
A
or f ¼ 2fnS , we can obtain a frequency
When f ¼ 2fm
peak as

Z
A
Þ¼
Sðk; 2fm

T

A

s2 ðk; tÞeÀj4pfm t dt

0

(11)

A

2 A
2 j2'm ðkÞ
' T ðbA
m Þ jam ðkÞj e

D' ¼ ½D'1 ; D'2 ; . . . ; D'L 

Z
Sðk; 2fnS Þ ¼

T

S

s2 ðk; tÞeÀj4pfn t dt

0

(12)

S

' T ðbSn Þ2 jaS0 ðkÞj2 ej2'n ðkÞ
whose value will be much higher than those of other frequency bins generated by the incoherent signal component
as observed by experiments. In (11) and (12), the approximation comes from the fact that, when the frequency is twice
that of the Doppler shift of a specific satellite (authentic one
or spoofing one), its energy will be accumulated coherently
thanks to the removal of the BPSK modulation, and hence, it
becomes a single-tone signal as was demonstrated in (9),
whereas the relative power of other satellite signals with different Doppler shifts and incoherent signal component will
be significantly reduced after integration.
After determining the frequency peaks that rise above
the noise floor by thresholding, all the authentic and spoofing satellites can be detected. Then, we can calculate the
intermediate phase difference between two antennas for
each detected satellite as
A
A
De
'A
m ¼ ﬀ½Sð2; 2fm Þ=Sð1; 2fm Þ

(13)

or
De
'S0 ¼ ﬀ½Sð2; 2fnS Þ=Sð1; 2fnS Þ:

(14)

Arising from the squaring process that doubles the
phase term, the relationship between the intermediate
phase difference De
' in (13) or (14) and the phase difference D' in (7) is given by


De
'=2; j2D'j p
De
'=2 À p; else:

(15)

Equation (15) means that directly using De
'=2 to estimate D' is ambiguous because, if j2D'j > p, then
we have De
'=2 ¼ ﬀej2½'ð2ÞÀ'ð1Þ =2 ¼ ﬀej2D' =2 ¼ D' þ p.
Therefore, in order to well suppress the spoofing signals,
ambiguity resolving is needed to get D' based on (15). However, for the spoofing detection purpose, this step is not necessary since the ambiguity is the same for all spoofing
satellites. In other words, we can simply use De
'=2 as a preliminary estimation of D' in the spoofing detection step.
MARCH 2021

(16)

where L is the number of detected satellites, i.e., the number of the determined frequency peaks.
Then, the successive differences of the phase-difference
vector D' can be computed to get the following vector:
DD' ¼ ½D'2 À D'1 ; . . . ; D'L À D'LÀ1 :

or

D' ¼

Based on this, we can get an ascending phase-difference vector as shown in the following equation by sorting the phase
differences of all the detected satellites:

(17)

Since all the spoofing satellites have the same D'S0 , the
spoofing phenomenon can be detected if the minimum of
DD' is smaller than a detection threshold, expressed as
min½DD'



(18)

where  is the spoofing detection threshold, which can be
set as a fixed value in advance or be determined by some
adaptive techniques in practical applications, e.g., the constant false-alarm rate (CFAR) detection technique [26].
We note that, for the proposed method, the spoofing
detection is conducted discretely with an interval of T . In
some cases, false detection alarms may be introduced by the
noise, the authentic satellites that come from near the same
direction, or the unexpected large phase differences among
different spoofing satellites. A backward sliding-average
based approach can be used to solve this problem, given by
DD'o ¼

À1
X
1 W
DD'oÀw
W w¼0

(19)

where W is the window size for average, DD'o corresponds to the oth spoofing detection point with time of
ðo À 1ÞT , and DD'oÀw corresponds to the (o À w)th detection point. After sliding-average along time, the negative
influences caused by the noise, authentic satellites from
the close directions, and unexpected spoofing satellite
phases can be reduced. Therefore, the spoofing detection
in (18) can be conducted based on DD' instead of DD'.
Furthermore, when there is spoofing, since the element
value of D' is continuously ascending, the spoofing satellites can also be distinguished from the authentic satellites
by determining an index set including continuous indexes
(from lfirst to llast ) of DD' with the corresponding elements
smaller than the detection threshold. To make it clear, with
a real-sampled GPS signal with spoofing, D' and DD' are
shown in Figure 3. It can be seen from the top subfigure
that, given lfirst ¼ 3 and llast ¼ 6, D'ðlfirst Þ; :::;
D'ðllast Þ; D'ðllast þ 1Þ are quite close to each other, which
is a strong sign of spoofing attack occurring. Based on DD'
as shown in the bottom subfigure, the spoofing phenomenon can be detected according to (18), whereas lfirst and
llast can also be determined to distinguish the spoofing
satellites.

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